Research Report

Acontinuum‐basedanalyticalapproachtoevaluateresponseofsinglepilesunder

dynamiclateralloading

Submitted to

Mid-Atlantic Universities Transportation Center

and

Larson Transportation Institute

The Pennsylvania State University, University Park

Nina Zabihi Dr. Prasenjit Basu

Dr. Swagata B. Basu

September 2014

2

Table of Contents

Abstract ........................................................................................................................................... 3

Introduction ..................................................................................................................................... 4

Problem definition .......................................................................................................................... 5

Displacement and strain fields .................................................................................................... 6

Governing differential equations ................................................................................................. 8

Solution algorithm ..................................................................................................................... 12

Solution for a specific case – undamped pile-soil system under steady state harmonic loading .. 14

Analytical Solution for Pile Deflection ..................................................................................... 15

Finite Difference Formulation for Soil Displacement Shape Functions ................................... 17

Results and Discussions ............................................................................................................ 20

Summary and Conclusions ........................................................................................................... 29

References ..................................................................................................................................... 30

3

Abstract

The pressing need for building resilient and sustainable civil infrastructure systems requires

optimal design of infrastructure components which are not only capable to perform adequately

under service loads but also resilient enough to survive under loads from man-made and natural

extreme events. This report focuses on a component of seismic fragility analysis of bridge

foundations – load-displacement behavior of piles under dynamic lateral loading. Winkler (soil-

spring) models are widely used to capture pile-soil interaction under lateral loading. However,

such simplified foundation-soil interaction model cannot predict ‘true’ serviceability and limit

conditions. Consequently, a high level of conservatism is employed in such analyses, and

therefore simplified soil spring models cannot be relied upon in the assessment of reliability (or

vulnerability) of bridge foundations under dynamic loading.

In order to effectively capture different possible damage conditions in foundations and their

complex interaction with structures, a continuum-based analytical framework is explored in this

study to investigate behavior of single piles under dynamic lateral loading. The governing

differential equations for the pile-soil system are derived by using Hamilton’s principle. The soil

displacement field is assumed to be consistent with two facts: (i) soil displacement decreases as

the distance from the pile increases and (ii) in addition to the radial distance from pile, soil

displacement at any point depends on the direction of the load with respect to that point. Because

of the interdependency between pile deflection and soil displacement, an iterative solution

scheme is adopted. The adopted analysis framework shows promise to eliminate the need for

computationally expensive numerical analyses (e.g., 3D finite element analysis). It is anticipated

that future research based on the work presented in this report will facilitate generation of pile

fragility curves that can provide probabilistic estimation of foundation vulnerability under

dynamic loading.

4

Introduction

Piles are often used as foundation elements for structures subjected to both axial and lateral

loads. In situations when structures are exposed to high lateral load demands (such as due to

machine vibrations, seismic motions, extreme wind load, sea waves, to name a few) response of

pile foundations under static and/or dynamic lateral load should be critically investigated to

avoid any catastrophic structural failure initiated by the failure of foundations.

Over the last few decades, different analytical and numerical approaches have been adapted

to model and analyze pile-soil interaction and dynamic response of pile foundations. Winkler

spring-based models are the simplest among all available models. In such models, soil

continuum and piles are, respectively, approximated as discretely-spaced and interconnected

springs. In some variations of such discrete modeling approach, dashpots are also connected to

one or more pile segments. The spring and dashpot constants are either back-calculated from

experimental data or directly fed in to the models based on analytical considerations and

judgement. The Novak model (Novak 1974) and the Matlock model (Matlock et al. 1978) are

known as the conventional Winkler models that are frequently used in dynamic response analysis

of pile foundations. The Nogami model (Nogami et al. 1988; 1991) is another variation of

spring-based model that can account for nonlinear pile-soil interaction under dynamic loading.

Other researchers also attempted to account for nonlinearity and inhomogeneity in the soil

continuum, pile-soil separation and hysteretic degradation of soil through modified versions of

Winkler model. For example, El Naggar and Bentley (2000) and Maheshwari and Watanaba

(2006) used nonlinear springs to take into account the gapping at the soil-pile interface.

Numerical simplicity and the ease of implementation are among the top advantages of Winkler

models; however, such models are not capable of (i) considering shear transfer within soil (i.e.,

shear interaction between two adjacent layers of soil) and (2) capturing the three-dimensional

interaction between the pile and the soil. Such shortcomings of Winkler models can be overcome

by using continuum-based methods.

Continuum-based approaches for analyzing pile-soil interaction under lateral loading can

broadly be divided into two categories: using numerical techniques (e.g., finite element, finite

difference) and using analytical or semi-analytical techniques. Finite element method has

extensively been used by researchers for analysis of laterally loaded piles (Kuhlemar 1979,

Krishna et al. 1983, Velez et al. 1983). In finite element analyses of piles under dynamic lateral

5

loading, the far field is represented by energy-absorbing boundaries. More advanced finite

element analyses consider the effects of soil plasticity and separation at pile-soil interface on

dynamic lateral response of piles (e.g. Bentley and El Naggar 2000, and Maheshwari et al. 2004).

In spite of their superior capability in analyzing soil-pile interaction, three-dimensional

numerical analyses are computationally expensive for routine practices. Therefore, analytical or

semi-analytical methods are more appealing.

Considering the soil surrounding the pile as a linear elastic continuum, Das et al. 1999

developed analytical solutions to obtain lateral pile deflection along the length of the pile.

However, they assumed the same decreasing trend of soil displacement in both the radial and

tangential directions with the increase of distance from the pile. In case of single pile under static

lateral loading, Basu et al. 2009 indicated that the consideration of the same decreasing trend of

soil displacement in both radial and tangential directions would result in a soil response stiffer

than that is in reality.

The present study explores a continuum-based, semi-analytical framework for dynamic

analysis of laterally loaded single piles. Pile and soil displacements under simultaneous actions

of dynamic lateral load F(t) and moment M(t) are quantified as a function of depth and time. The

analytical formulation is based on the facts that soil displacement decreases as the distance from

the pile increases and displacement at any point in the soil surrounding the pile depends on the

direction of applied load with respect to the point of interest. Hamilton’s principle is utilized in

deriving the governing differential equations that describe pile and soil displacements.

Appropriate boundary conditions are enforced to obtain simultaneous solution for pile and soil

displacements. For a specific case of steady state harmonic loading, closed-form solution is

obtained for pile deflection. Soil displacement fields are evaluated using one-dimensional finite

difference technique.

Problem definition

We consider a circular pile with length Lp and radius rp embedded in a semi-infinite,

homogeneous, isotropic soil deposit. The pile head is subjected to a time dependent lateral force

F(t), and a time dependent moment M(t), such that F(t) and M(t) are orthogonal vectors (Fig. 1).

The cylindrical coordinate system (r, , z) is employed with its origin at the center of the pile

6

head and the positive z axis (which coincides with the pile axis) pointing downwards. Specific

assumptions made in the present analysis are:

i. constitutive behavior of soil is elastic with shear modulus Gs and Lame’s constant λs

ii. the pile is vertical and it behaves as an Euler-Bernouli beam with a constant flexural

rigidity EpIp throughout its length; Ep and Ip are, respectively, the Young’s modulus of

pile material and second moment of inertia for pile cross section

iii. the pile is perfectly connected to the soil; i.e. there is no separation or slippage at the pile

and soil interface

iv. vertical displacement of the pile under the lateral load and moment is negligible.

v. time-dependent pile displacement is a function of depth z only; i.e., displacement at every

point on a pile cross section at depth z is constant

Figure 1 Problem geometry – single pile under dynamic lateral loading

Displacement and strain fields

Displacement of any point at depth z within the ground is assumed to be a function of radial pile

displacement w(z, t) at the same depth (Fig. 2). Dimensionless displacement shape functions

ϕ r and ϕ r are used to account for the decay in soil displacement with increase in radial

distance away from the pile. ϕ r and ϕ r are equal to 1 at r r (compatibility at pile and

soil interface) and are equal to 0 at r → ∞ . For the domain below the pile (i.e., for z L ),

7

w(z, t) is the displacement of the soil column, with radius rp and extending infinitely below the

pile. Displacement components at any point within the soil domain are expressed as:

u r, θ, z, t w z, t ϕ r cos θ (1a)

u r, θ, z, t w z, t ϕ r sin θ (1b)

u z, t, r, θ 0 (1c)

Figure 2 Soil displacement components at a point within the ground

The strain components at any point within the ground are derived following the displacement

components described in equation (1).

εεεγγγ

w z cos θ

w z cos θ

0w z sin θ

ϕ r cos θ

ϕ r sin θ

(2)

8

Governing differential equations

The governing differential equations for pile deflection and soil displacement are obtained

following the Hamilton’s principle for deformable bodies. Mathematically,

δW δT δU dt 0 (3)

where Wnc is work done by non-conservative forces, T and U are respectively, kinetic and

potential energy associated with the pile-soil system. The operator is used to signify variation

of a physical quantity. The potential energy U of the pile and soil system is given by:

U E I dz σ ε σ ε τ γ τ γ

τ γ rdrdθdz σ ε σ ε τ γ τ γ

τ γ rdrdθdz (4)

Combining equations (2) and (4), the potential energy of the pile-soil system can be rewritten as:

U E I dz λ 2G w 2λ w

λ 3G w G w 2G w G ϕ

G ϕ rdrdz r G dz (5)

The kinetic energy T of the pile and soil system is:

T ρ A dz ρ A dz ρ

rdzdrdθ (6)

where ρ and ρ are density of the pile materials and soil, respectively, and A is cross section

area of the pile. Using equation (1) in (6), kinetic energy T for the pile-soil system can be

rewritten as:

9

T ρ A dz ρ A dz ρ ϕr

ϕθ rdrdz (7)

Now, the work done by the non-conservative forces in variational form can be expressed as:

W F t . δw 0, t M t , c δwdz (8)

where, c represents damping of the pile-soil system. Combining equations (3), (5), (7), and (8)

and applying principles of variational calculus:

δW δT δU dt A w δw B w δ C ϕr δϕr

D ϕθ δϕθ dt 0 (9)

The terms associated with each of the variations δw, δ , δϕ , and δϕ must individually be

equal to zero to satisfy equation (9). This exercise yields the governing differential equations and

boundary conditions needed for solving w(z,t), ϕ r , and ϕ r .

Collecting the coefficients of δw and δ for the domains 0 z L and L z and

equating each to zero yields the differential equations and boundary conditions that govern the

radial deflections of the pile and the soil column (extending infinitely) below the pile.

Differential equations

E I 2τ kw M c 0 for 0 z L (10)

2τ kw M c 0 for z L (11)

Boundary conditions

At the ground surface (i.e. z 0 ):

E I 2τ F (12a)

10

E I M (12b)

At the bottom of the pile (i.e.z L ):

E I 2τ 2τ (13a)

E I 0 (13b)

Pile-soil interface at z L :

w w (14a)

At z → ∞ (i.e., at a distance far below the pile base):

w 0 (14b)

Parameters M , M , τ, τ , and k in equations (10) through (13) are defined as:

M ρ A πρ r ϕ ϕ dr (15a)

M ρ A πρ r ϕ ϕ dr (15b)

2τ πG r ϕ ϕ dr (15c)

2τ 2τ πGr (15d)

k π λ 2G η Gη 2λη 2Gη 2λη 2Gη λ 3G η

λ 3G η 2 λ 3G η (15e)

The terms in equation (15c) are functions of displacement shape functions and their

derivatives.

11

ηdϕdr

rdr ηdϕdr

rdr

η ϕdϕdr

dr η ϕdϕdr

dr

η ϕdϕdr

dr η ϕdϕdr

dr (16)

ηϕrdr η

ϕrdr

ηϕ ϕr

dr

The differential equations to calculate soil displacement shape functions ϕ r and ϕ r are

obtained by collecting the terms containing δϕ , and δϕ and equating each to zero. The coupled

differential equations describing ϕ and ϕ are:

ϕ ϕ (17)

ϕ ϕ (18)

Equations (17) and (18) are subjected to boundary conditions ϕ ϕ 1 at r r and

ϕ ϕ 1 at r → ∞. The parameters γ , γ , γ , and γ in equations (17) and (18) are

constants depending on soil properties and γ and γ depend on w(z,t) at any given time and

depth. These parameters are defined as:

γmm

1G

λ 2G

γr

n Lm

γm m

mλ Gλ 2G

γmm

3λG

(19)

γr

n Lm

γm mm

1λG

12

where:

m λ 2G w dz m G w

m λ w dz m λ 3G w dz (20)

n G∂w∂z

L ρ∂w∂t

Solution algorithm

In order to obtain pile deflection as a function of depth and time, equation (10) should be solved.

However, solution of equation (10) needs equation (11) to be solved first in order to satisfy

boundary condition specified in equation (13a). Moreover, the coefficients k, M1, M2, τ and τ in

equations (10) and (11) depend on ϕ and ϕ , which are not known a priori. ϕ and ϕ , on the

other hand, are dependent on w(z,t) and its derivatives through the parameters γ and γ (see

equations 17, 18, and 19). Therefore, an iterative algorithm is necessary to solve the problem.

Note that equations (17) and (18) are interdependent and should be solved simultaneously. So an

iterative solution procedure is also warranted in order to quantify soil displacement shape

functions. The flow chart presented in Fig. 3 shows the iterative solution steps.

13

(a)

(b)

Fig. 3. Solution flow chart: (a) for finding w(z,t), (b) for finding ϕ r and ϕ r

14

Solution for a specific case – undamped pile-soil system under steady state harmonic loading

The analytical framework outlined in the previous sections is used to solve for a specific case of

undamped pile-soil system (i.e., c=0) subjected to steady state harmonic loading. The harmonic

loading functions considered for this case are:

F t F e (21a)

M t M e (21b)

where Ω is the circular loading frequency and F and M are the amplitudes of the lateral load

F(t) and the moment M(t), respectively. Pile and soil displacement are assumed to be in-phase

with the loading function. Therefore, the motion of the pile and the soil column beneath it will be

in the form of:

w z, tw z e ,0 z L

w z e ,z L (22)

where w z is the amplitude of the pile displacement, and w z is the displacement amplitude

for the soil column just beneath the pile. Substitution of equations (21) and (22) into equations

(10) and (11) yields:

Differential equations

E I 2τ k M Ω w 0 for 0 z L (23)

2τ k M Ω w 0 for z L (24)

Boundary conditions

At ground surface (i.e. z 0 ):

E I 2τ F (25a)

15

E I M (25b)

At pile base (i.e.z L ):

E I 2τ 2τ 0 (26a)

E I 0 (26b)

At the interface of soil and pile (i.e. z L ):

w w (27a)

At a point far below the pile; i.e., for z → ∞

w 0 (27b)

The solution of equation (24) with boundary conditions expressed through equations (27a)

and (27b) is given by:

w z w L e (28)

Equation (28) is used in equation (26a) to obtain the pile deflection.

Analytical Solution for Pile Deflection

The general solution of equation (23) can be written as:

w z C w C w C w C w (29)

where w , w , w , and w are individual solutions of the fourth order differential equation

and C , C , C , and C are integration constants. w , w , w , and w are trigonometric or

hyperbolic functions arising in the solution of linear ordinary differential equations. Finding the

individual solutions using Table 1, and then applying the boundary conditions specified in

16

equations (25) and(26) into equation (29), the integration constants, C , C , C , and C , can be

determined. Equation (23) can be rearranged as:

2B Aw 0 (30)

where the two parameters A and B are expressed as:

A (31a)

B (31b)

Based on the relative magnitude of A and B2, two cases are considered here. For each case,

two other parameters, a and b, are defined and those are used in finding the individual solutions

of the differential equation:

Case 1: A B

√A B (32a)

b √A B (32b)

Case 1: A B

B √B A (33a)

b B √B A (33b)

The individual solutions of equation (23), represented through equation (30), and

their derivatives are outlined in Table 1. However, soil displacement shape functions must be

quantified first to calculate individual solutions wp1 through wp4.

17

Table 1Individual solutions for pile displacement in equation 29

B A

Individual

solutions

Functions and their derivatives

w w w w

w sinh azcosh bz aw bw a b w

2ab w

a a 3b w

b b 3a w

w cosh azcosh bz aw bw a b w

2ab w

a a 3b w

b b 3a w

w cosh azsinh bz aw bw a b w

2ab w

a a 3b w

b b 3a w

w sinh azsinh bz aw bw a b w

2ab w

a a 3b w

b b 3a w

A B

Individual

solutions

Functions and their derivatives

w w w w

w sinh az aw a w a w

w cosh az aw a w a w

w sinh bz bw b w b w

w cosh bz bw a w b w

Finite Difference Formulation for Soil Displacement Shape Functions

The differential equations for ϕ and ϕ (equations 17 and 18) are solved numerically using

finite difference formulation. Using the central-difference scheme, finite difference forms of

equations (17) and (18) can be written as:

18

ϕ

ϕ (34)

ϕ

ϕ (35)

where Δr is discretization length and j is the nodal index (Fig. 4).

Figure 4 Finite difference discretization in radial direction

Δr should be sufficiently small to maintain a satisfactory level of accuracy and the total number

of nodes m should be sufficiently large to adequately model the infinite domain in the radial

direction,. After applying the boundary conditions, equation (34) and (35) can be written in

matrix form as:

K∅ ∅ F∅ (36a)

K∅ ∅ F∅ (36b)

In equation (36), K∅ and K∅ are tri-diagonal matrices with elements K∅ and K∅

rj

1 2 3 4 5 j j+1 m-1 m

∆r

j-1

1≤ j ≤m

19

K∅

1,i j 1orm

∆ ∆,i j 1 andj 2

∆,2 i j m 1

∆ ∆ , i j 1andj m 1

0,others

(37a)

K∅

1,i j 1orm

∆ ∆,i j 1 andj 2

∆,2 i j m 1

∆ ∆ , i j 1andj m 1

0,others

(37b)

and the elements of F∅ and F∅ are:

F∅

1,j 1

∆ ∆ ϕ ,j 2

ϕ ,3 j m 2

ϕ ,j m 1

0,j m

(38a)

F∅

1,j 1

∆ ∆ ϕ ,j 1

ϕ ,3 j m 2

ϕ ,j m 1

0,j m

(38b)

20

Since, the right-hand side vectors F∅ and F∅ in equation (36) have elements

containing the unknown ϕ and ϕ , iterative procedures should be followed to obtain their

values as illustrated in solution flowchart (Fig.3b). A MATLAB code is developed to handle the

iterative solution procedure described herein. Note that the CPU runtime of the MATLAB code

(run on a core-i5 processor with 8 GB RAM) is approximately 100 s for the example cases

presented in this report.

Results and Discussions

The application of the developed analysis methodology is demonstrated using an example

problem with input parameters listed in Table 2.

Table 2- Input parameters used in the analysis of sample problem

Pile Geometry Radius, rp (mm) 0.5

Length, Lp (m) 5, 10, 15, 20

Material Properties for Pile Modulus of Elasticity, E (Gpa) 25

Density, ρ (kg/m3) 2400

Material Properties for Soil

Modulus of Elasticity, E (Mpa) 25

Poisson Ratio,ν 0.3

Density, ρ (kg/m3) 1500

Force Amplitude, F0 (kN) 1000

Circular Frequency, Ω (rad/s) 10

The effects of pile length Lp and pile-to-soil stiffness ratio are also investigated. Fig. 5 shows the

amplitude of pile deflection, , for different pile lengths (= 5m, 10m, 15m, and 20m). For

pile length Lp = 5m, a rigid body rotation that signifies short-pile response under lateral loading,

is observed. A transition behavior is observed for Lp = 10m and the pile behaves as a long-pile

for Lp = 15 and 20m. Figure 6 shows, at some selected time instants, the variation of pile

deflection with depth.

21

a) b)

c) d)

Fig.5. Amplitude of pile deflection along the pile length (wp(z))

0

1

2

3

4

5

6

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Dep

th, z

(m

)

Amplitude of Pile Deflection (mm)

Lp= 5m

0

2

4

6

8

10

12

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Dep

th, z

(m

)

Amplitude of Pile Deflection (mm)

Lp=10 m

0

2

4

6

8

10

12

14

16

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Dep

th, z

(m

)

Amplitude of Pile Deflection (mm)

Lp=15 m

0

2

4

6

8

10

12

14

16

18

20

22

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Dep

th, z

(m

)

Amplitude of Pile Deflection (mm)

Lp=20 m

22

Fig.6. Pile Deflection at different time instants

The bending moment and shear force at different pile cross-sections are important design

parameters for laterally loaded piles. Figure 7 shows the variation of bending moment and shear

force with along the length of a 10-m-long pile. The maximum bending moment occurs at a

depth of 2 m (= Lp/10) from the pile head and the maximum shear force occurs at the pile head.

7(a)

0

2

4

6

8

10

12

14

16

18

20

22

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Dep

th, z

(m

)

Pile Deflection (mm)

t=0.01 s

t=0.1 s

t=0.15 s

t=0.2 s

t=0.25 s

t=0.3 s

Lp=20 m

0

5

10

15

20

-50 0 50 100 150 200 250 300 350 400 450 500 550 600

Dep

th, z

(m

)

Bendind moment (kN.m)

23

7(b).

Fig.7. (a) Bending moment and (b) Shear force diagram along the pile length

A parametric study is done to investigate the effects of parameters Lp/rp (i.e., slenderness ratio of

the pile) and Ep/Es (i.e., pile-soil modulus ratio) on pile behavior under harmonic lateral loading.

Figure 8 shows that lateral displacement of pile base is more sensitive to change in pile

slenderness ratio Lp/rp when compared to pile head displacement. Both pile head and base

displacements decreases with increase in Lp/rp; nonetheless, they become insensitive to the

change in slenderness ratio for Lp/rp ≥ 25.

Fig.8. Effect of pile slenderness ratio Lp/rp on pile head and base displacements

0

5

10

15

20

-200 0 200 400 600 800 1000 1200

Dep

th, z

(m)

Shear force (kN)

-7-6-5-4-3-2-101234567

0 10 20 30 40 50

Pil

e di

spla

cem

ent,

wp(

z) (

mm

)

Lp/rp

rp=0.5 m Pile head displacement, wp at z=0

Pile base displacement, wp at z=Lp

24

For a long pile (Lp = 25 m), the ratio Ep/Es of elastic modulus of deformation for pile and soil

significantly affects the displacement at the pile head; pile head displacement appears to be

linearly proportional with Ep/Es. The displacement at pile base also increases as the ratio Ep/Es

increases ( or Es decreases); however, the relationship between these two parameters is not linear

(Fig. 9).

Fig. 9. Effect of Ep/Es on the displacement of the pile at the head and the bottom

The variations of soil displacement shape functions, ϕ and ϕ , with radial distance from the

pile are presented in Figure 10. The slope of decay in the soil displacement shape function is

larger in tangential direction than that in the radial direction. Figures 11 and 12 shows soil

displacements ur and u recorded at different depths z and at different angular coordinates

θ 0, , , ) with respect to the direction of the applied force. Note that ur = u = 0 for θ 0

and θ , which directly follows the assumption of no separation between pile and soil

surrounding it (an assumption necessary for the present analytical continuum-based approach).

Figure 13 demonstrates the effects of radial distance and angle with respect to the direction of the

applied load on the radial and tangential displacement at any point (for depth z = 0.5 m).

-40

-20

0

20

40

60

80

100

120

0 2000 4000 6000 8000 10000 12000

Pil

e di

spla

cem

ent,

wp(

z) (

mm

)

Ep/Es

Pile head displacement(wp at z=0)

Pile base displacement(wp at z=Lp)

Lp=20 mEp=25 Gpa

25

Fig. 10. variation of soil displacement shape functions, ϕ r and ϕ r

11(a)

0

0.2

0.4

0.6

0.8

1

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t sha

pe f

ucti

ons,

ϕ r

, ϕ ϴ

Normalized radial distance from the pile centerline, r/rp

ϕr

ϕϴ

-1

0

1

2

3

4

5

6

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t in

radi

al d

irec

tion

, ur(m

m)

Normalized distance from the pile centerline, r/rp

ϴ=0

z=0.5 m

z=3 m

z=8 m

z=20 m

z=25 m

F(t)

ur

26

11(b)

11(c)

Fig.11. Soil displacement in radial direction at the angel of (a) θ 0,(b) θ , (c)θ with

respect to the direction of the applied force, at different depth, (Lp=20 m)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 6 11 16 21 26 31 36

Soil

disp

lace

men

t in

radi

al d

irec

tion,

ur(m

m)

Normalized distance from the pile centerline, r/rp

ϴ=П/6

z=0.5 m

z=3 m

z=8 m

z=20 m

z=25 m

F(t)

ur

-0.5

0

0.5

1

1.5

2

2.5

3

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t in

radi

al d

irec

tion

, ur(m

m)

Normalized distance from the pile centerline, r/rp

ϴ=П/3

z=0.5 m

z=3 m

z=8 m

z=20 m

z=25 m

F(t)

ur

27

12(a)

12(b)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1 6 11 16 21 26 31 36So

il di

spla

cem

ent i

n ta

ngen

tial d

irec

tion,

u ϴ(m

m)

Normalized distance from the pile centerline, r/rp

ϴ=П/6

z=0.5 m

z=3 m

z=8 m

z=20 m

z=25 m

F(t)

uϴ

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t in

tang

entia

l dir

ectio

n,u ϴ

(mm

)

Normalized distance from the pile centerline, r/rp

ϴ=П/3

z=0.5 m

z=3 m

z=8 m

z=20 m

z=25 m

F(t)

uϴ

28

12(c)

Fig.12. Soil displacement in tangential direction at the angel of (a) θ ,(b)θ , (c) θ

with respect to the direction of the applied force, at different depth, (Lp=20 m)

13(a)

-6

-5

-4

-3

-2

-1

0

1

1 6 11 16 21 26 31 36So

il di

spla

cem

ent i

n ta

ngen

tial d

irec

tion,

u ϴ(m

m)

Normalized distance from the pile centerline, r/rp

ϴ=П/2

z=0.5 m

z=3 m

z=8 m

z=20 mz=25 m

F(t)

uϴ

0

1

2

3

4

5

6

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t in

radi

al d

irec

tion,

ur

(mm

)

Normalized distance from the pile centerline, r/rp

z=0.5 m

a

b

c

d

F(t)a: ϴ=0b: ϴ=П/6c: ϴ=П/3d: ϴ=П/2

F(t)F(t)F(t) a

bcd

29

13(b)

Fig.13. Variation of soil displacement (for Lp=20 m and z=0.5 m) at various locations within the domain surrounding the pile (a) radial displacement ur and (b) tangential displacement u

Summary and Conclusions

A semi-analytical continuum-based approach is developed for predicting response of a single

pile subjected to dynamic lateral loading. Soil surrounding the pile is considered to be elastic,

homogeneous and isotropic. A MATLAB code is developed to perform analysis following the

proposed framework. It appears that the developed semi-analytical framework is computationally

efficient (much so when compared to 3D FEAs). Although results show promise of the analysis

methodology presented in this report, accuracy of such results needs further verification using

real-lie data and/or conventional three-dimensional finite element analyses.

-6

-5

-4

-3

-2

-1

0

1

1 6 11 16 21 26 31 36

Soi

l dis

plac

emen

t in

tang

entia

l dir

ectio

n,u ϴ

(mm

)

Normalized distance from the pile centerline, r/rp

z=0.5 m

a

b

c

d

F(t)a: ϴ=0b: ϴ=П/6c: ϴ=П/3d: ϴ=П/2

F(t)F(t)F(t) a

bcd

30

References

Basu, D., Salgado, R., Prezzi, M., 2009, A Continuum-Based Model for Analysis of Laterally

Loaded Piles in Layered Soils, Geotechnique 59(2), 127-140

Bentley, K. J., El Naggar, M. H., 2000, Numerical Analysis of Kinematic response of Single

Piles, Can. Geotech. J., 37(6), 1368-1382

Das, Y. C., Sargand, S. M., 1999, Forced Vibration of Laterally Loaded Piles, international

journal of solids and structures, 36, 4975-4989

El nagger, M. H., Bentley, K. J., 2000, Dynamic Analysis for Laterally Loaded Piles and

Dynamic p-y Curves, Can. Geotech. J., 37(6), 1166-1183

Krishna, R., Gazetes, G., Velez, A., 1983, Static and Dynamic lateral deflection of piles in

nonhomogeneous soil stratum, Geotechnique,33(3), 307-326

Kuhlemar, R. L., 1979, Static and Dynamic laterally loaded floating piles, J. Geotech. Engrg.,

105(2) 289-304

Maheshwari, B. K., Truman, K. Z., El Naggar, M. H., Gould, P. L., 2004, Three-Dimensional

Finite Element Nonlinear Dynamic Analysis ofPile Groups for Lateral Transient and

Seismic Excitations, Can. Geotech. J., 41(1), 118-133

Maheshwari, B. K., and Watanabe, H., 2006, Nonlinear Dynamic Behavior of Pile Foundations:

Effects of Separation at Soil-Pile Interface, Soils Found., 46(4), pp. 437-448

Matlock, H., Foo, S. H., Bryant, L. L. 1978, Simulation of Lateral pile Behavior, Proc.

Earthquake Engrg. And Soil Dyn. ASCE, Pasadena, California, July, 600-619

Nogami, T., Konagai, K., 1988, Time-Domain Flexural Response of Dynamically Loaded Single

Piles, Engrg. Mech. ASCE, 114(9), 1512-1525.

Nogami, T., Otani, J., Konagai, K., Chen, H.L., 1991, Nonlinear Soil-Pile Interaction Model for

Dynamic Lateral Motion, J. Geotech. Engrg, ASCE

Novak, M., 1974, Dynamic Stiffness and Damping of Soils, J. Can. Geotech. Engrg. NRC of

Canada, 11(5), 574-698.

Velez, A., Gazetas, G., Krishna, R., 1983, Lateral Dynamic Response of Constrained-Head Piles,

ASCE J. Geotech. Engrg., 109(8), 1063-1081